3.1 | The Derivative and the Tangent Line Problem


If \(f\) is defined on an open interval containing \(c\), and if the limit
$$
\lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} = \lim_{x \rightarrow 0} \frac{f(c+x)-f(c)}{\Delta x} = m
$$
exists, then the line passing through \((c,f(c))\) with slope \(m\) is the tangent line to the graph of \(f\) at the point \((c, f(c))\).
The derivative of \(f\) at \(x\) is
$$
f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x) – f(x)}{\Delta x}
$$
provided the limit exists. For all \(x\) for which this limit exists, \(f’\) is a function of \(x\).
If \(f\) is differentiable at \(x = c\), then \(f\) is continuous at \(x = c\).

E 3.1 Exercises